If $\frac{a}{b} = \frac{1}{3}$, $\frac{b}{c} = 2$, $\frac{c}{d} = \frac{1}{2}$, $\frac{d}{e} = 3$ and $\frac{e}{f} = \frac{1}{4}$, then what is the value of $\frac{abc}{def}$?

### Concept & Formula This problem involves finding a composite ratio by strategically grouping the given individual ratios. Rather than finding individual values for each variable, you can regroup the target expression $\frac{abc}{def}$ into a product of chained ratios. $$ \frac{abc}{def} = \left(\frac{a}{d}\right) \times \left(\frac{b}{e}\right) \times \left(\frac{c}{f}\right) $$ ### Step-by-Step Solution * **Given:** * Ratios: $\frac{a}{b}=\frac{1}{3}$, $\frac{b}{c}=2$, $\frac{c}{d}=\frac{1}{2}$, $\frac{d}{e}=3$, $\frac{e}{f}=\frac{1}{4}$ * Target expression: $\frac{abc}{def}$ * **Calculation:** 1. Rewrite the target expression into chained groups: $\left(\frac{a}{d}\right) \times \left(\frac{b}{e}\right) \times \left(\frac{c}{f}\right)$. 2. Calculate $\frac{a}{d}$ by multiplying the sequential ratios from $a$ to $d$: $\frac{a}{d} = \left(\frac{a}{b}\right) \times \left(\frac{b}{c}\right) \times \left(\frac{c}{d}\right) = \left(\frac{1}{3}\right) \times (2) \times \left(\frac{1}{2}\right) = \frac{1}{3}$. 3. Calculate $\frac{b}{e}$ by multiplying the sequential ratios from $b$ to $e$: $\frac{b}{e} = \left(\frac{b}{c}\right) \times \left(\frac{c}{d}\right) \times \left(\frac{d}{e}\right) = (2) \times \left(\frac{1}{2}\right) \times (3) = 3$. 4. Calculate $\frac{c}{f}$ by multiplying the sequential ratios from $c$ to $f$: $\frac{c}{f} = \left(\frac{c}{d}\right) \times \left(\frac{d}{e}\right) \times \left(\frac{e}{f}\right) = \left(\frac{1}{2}\right) \times (3) \times \left(\frac{1}{4}\right) = \frac{3}{8}$. 5. Substitute these grouped values back into the rearranged target expression: $\frac{abc}{def} = \left(\frac{1}{3}\right) \times (3) \times \left(\frac{3}{8}\right)$. 6. Simplify the product: The $3$s cancel out, leaving $\frac{3}{8}$. ### Exam Strategy & Shortcut An even faster way is to recognize that $\frac{abc}{def}$ can be expressed simply by substituting proportional values if we create a chain. However, multiplying the sequences as shown above is the most robust and error-free method. Grouping terms into $\frac{a}{d}$, $\frac{b}{e}$, and $\frac{c}{f}$ perfectly aligns with the given chain. ### Common Pitfall A common mistake is trying to define a single variable, like $f$, as a common base (e.g., $e = \frac{f}{4}$, $d = \frac{3f}{4}$, etc.) and expressing every other variable in terms of $f$. This creates an enormous, tangled fraction that is extremely prone to arithmetic errors. ### Final Answer Therefore, the correct answer is $\frac{3}{8}$.